def test_laguerre():
assert laguerre(0, x) == 1
assert laguerre(1, x) == -x + 1
assert laguerre(2, x) == x**2 / 2 - 2 * x + 1
assert laguerre(3, x) == -x**3 / 6 + 3 * x**2 / 2 - 3 * x + 1
assert laguerre(n, oo) == (-1)**n * oo
assert laguerre(n, -oo) == oo
X = laguerre(n, x)
assert isinstance(X, laguerre)
assert laguerre(n, 0) == 1
assert conjugate(laguerre(n, x)) == laguerre(n, conjugate(x))
assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
pytest.raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(1))
pytest.raises(ValueError, lambda: laguerre(-2.1, x))
# issue sympy/sympy#10961
X = laguerre(Rational(5, 2), x)
assert isinstance(X, laguerre)
def test_laguerre():
assert laguerre(0, x) == 1
assert laguerre(1, x) == -x + 1
assert laguerre(2, x) == x**2/2 - 2*x + 1
assert laguerre(3, x) == -x**3/6 + 3*x**2/2 - 3*x + 1
assert laguerre(n, oo) == (-1)**n*oo
assert laguerre(n, -oo) == oo
assert laguerre(-n, x) == exp(x)*laguerre(n - 1, -x)
X = laguerre(n, x)
assert isinstance(X, laguerre)
assert laguerre(n, 0) == 1
assert conjugate(laguerre(n, x)) == laguerre(n, conjugate(x))
assert diff(laguerre(n, x), x) == -assoc_laguerre(n - 1, 1, x)
pytest.raises(ArgumentIndexError, lambda: laguerre(n, x).fdiff(1))
pytest.raises(ValueError, lambda: laguerre(-2.1, x))
# issue sympy/sympy#10961
X = laguerre(Rational(5, 2), x)
assert isinstance(X, laguerre)
def test_assoc_legendre():
Plm = assoc_legendre
Q = sqrt(1 - x**2)
assert Plm(0, 0, x) == 1
assert Plm(1, 0, x) == x
assert Plm(1, 1, x) == -Q
assert Plm(2, 0, x) == (3*x**2 - 1)/2
assert Plm(2, 1, x) == -3*x*Q
assert Plm(2, 2, x) == 3*Q**2
assert Plm(3, 0, x) == (5*x**3 - 3*x)/2
assert Plm(3, 1, x).expand() == ((3*(1 - 5*x**2)/2).expand()*Q).expand()
assert Plm(3, 2, x) == 15*x * Q**2
assert Plm(3, 3, x) == -15 * Q**3
# negative m
assert Plm(1, -1, x) == -Plm(1, 1, x)/2
assert Plm(2, -2, x) == Plm(2, 2, x)/24
assert Plm(2, -1, x) == -Plm(2, 1, x)/6
assert Plm(3, -3, x) == -Plm(3, 3, x)/720
assert Plm(3, -2, x) == Plm(3, 2, x)/120
assert Plm(3, -1, x) == -Plm(3, 1, x)/12
X = Plm(n, m, x)
assert isinstance(X, assoc_legendre)
assert Plm(n, 0, x) == legendre(n, x)
assert Plm(n, m, 0) == 2**m*sqrt(pi)/(gamma((1 - m - n)/2)*gamma(1 - (m - n)/2))
pytest.raises(ValueError, lambda: Plm(-1, 0, x))
pytest.raises(ValueError, lambda: Plm(0, 1, x))
pytest.raises(ValueError, lambda: Plm(-1, 2, x))
assert conjugate(assoc_legendre(n, m, x)) == \
assoc_legendre(n, conjugate(m), conjugate(x))
assert assoc_legendre(n, m, x).diff(x) == \
(n*x*assoc_legendre(n, m, x) -
(m + n)*assoc_legendre(n - 1, m, x))/(x**2 - 1)
pytest.raises(ArgumentIndexError, lambda: assoc_legendre(n, m, x).fdiff(1))
assert (assoc_laguerre(n, m, x).diff(m) ==
Sum(assoc_laguerre(k, m, x)/(-m + n), (k, 0, n - 1)))
def test_assoc_legendre():
Plm = assoc_legendre
Q = sqrt(1 - x**2)
assert Plm(0, 0, x) == 1
assert Plm(1, 0, x) == x
assert Plm(1, 1, x) == -Q
assert Plm(2, 0, x) == (3*x**2 - 1)/2
assert Plm(2, 1, x) == -3*x*Q
assert Plm(2, 2, x) == 3*Q**2
assert Plm(3, 0, x) == (5*x**3 - 3*x)/2
assert Plm(3, 1, x).expand() == (( 3*(1 - 5*x**2)/2 ).expand() * Q).expand()
assert Plm(3, 2, x) == 15*x * Q**2
assert Plm(3, 3, x) == -15 * Q**3
# negative m
assert Plm(1, -1, x) == -Plm(1, 1, x)/2
assert Plm(2, -2, x) == Plm(2, 2, x)/24
assert Plm(2, -1, x) == -Plm(2, 1, x)/6
assert Plm(3, -3, x) == -Plm(3, 3, x)/720
assert Plm(3, -2, x) == Plm(3, 2, x)/120
assert Plm(3, -1, x) == -Plm(3, 1, x)/12
X = Plm(n, m, x)
assert isinstance(X, assoc_legendre)
assert Plm(n, 0, x) == legendre(n, x)
assert Plm(n, m, 0) == 2**m*sqrt(pi)/(gamma((1 - m - n)/2)*gamma(1 - (m - n)/2))
pytest.raises(ValueError, lambda: Plm(-1, 0, x))
pytest.raises(ValueError, lambda: Plm(0, 1, x))
pytest.raises(ValueError, lambda: Plm(-1, 2, x))
assert conjugate(assoc_legendre(n, m, x)) == \
assoc_legendre(n, conjugate(m), conjugate(x))
assert assoc_legendre(n, m, x).diff(x) == \
(n*x*assoc_legendre(n, m, x) -
(m + n)*assoc_legendre(n - 1, m, x))/(x**2 - 1)
pytest.raises(ArgumentIndexError, lambda: assoc_legendre(n, m, x).fdiff(1))
assert (str(assoc_laguerre(n, m, x).diff(m)) ==
'Sum(assoc_laguerre(_k, m, x)/(-m + n), (_k, 0, n - 1))')
def test_laguerre_2():
# This fails due to issue for Sum, like issue sympy/sympy#5539
assert diff(assoc_laguerre(n, alpha, x), alpha) == \
Sum(assoc_laguerre(k, alpha, x)/(-alpha + n), (k, 0, n - 1))
def test_assoc_laguerre():
# generalized Laguerre polynomials:
assert assoc_laguerre(0, alpha, x) == 1
assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
assert assoc_laguerre(2, alpha, x).expand() == \
(x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
assert assoc_laguerre(3, alpha, x).expand() == \
(-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
(alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
# Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
X = assoc_laguerre(n, m, x)
assert isinstance(X, assoc_laguerre)
assert assoc_laguerre(n, 0, x) == laguerre(n, x)
assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
p = Symbol('p', positive=True)
assert assoc_laguerre(p, alpha, oo) == (-1)**p * oo
assert assoc_laguerre(p, alpha, -oo) == oo
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
pytest.raises(ArgumentIndexError, lambda: assoc_laguerre(n, m, x).fdiff(1))
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
pytest.raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
# issue sympy/sympy#10961
X = assoc_laguerre(Rational(5, 2), alpha, x)
assert isinstance(X, assoc_laguerre)
def test_laguerre_2():
# This fails due to issue for Sum, like issue sympy/sympy#5539
assert diff(assoc_laguerre(n, alpha, x), alpha) == \
Sum(assoc_laguerre(k, alpha, x)/(-alpha + n), (k, 0, n - 1))
def test_assoc_laguerre():
# generalized Laguerre polynomials:
assert assoc_laguerre(0, alpha, x) == 1
assert assoc_laguerre(1, alpha, x) == -x + alpha + 1
assert assoc_laguerre(2, alpha, x).expand() == \
(x**2/2 - (alpha + 2)*x + (alpha + 2)*(alpha + 1)/2).expand()
assert assoc_laguerre(3, alpha, x).expand() == \
(-x**3/6 + (alpha + 3)*x**2/2 - (alpha + 2)*(alpha + 3)*x/2 +
(alpha + 1)*(alpha + 2)*(alpha + 3)/6).expand()
# Test the lowest 10 polynomials with laguerre_poly, to make sure it works:
for i in range(10):
assert assoc_laguerre(i, 0, x).expand() == laguerre_poly(i, x)
X = assoc_laguerre(n, m, x)
assert isinstance(X, assoc_laguerre)
assert assoc_laguerre(n, 0, x) == laguerre(n, x)
assert assoc_laguerre(n, alpha, 0) == binomial(alpha + n, alpha)
p = Symbol('p', positive=True)
assert assoc_laguerre(p, alpha, oo) == (-1)**p*oo
assert assoc_laguerre(p, alpha, -oo) == oo
assert diff(assoc_laguerre(n, alpha, x), x) == \
-assoc_laguerre(n - 1, alpha + 1, x)
pytest.raises(ArgumentIndexError, lambda: assoc_laguerre(n, m, x).fdiff(1))
assert conjugate(assoc_laguerre(n, alpha, x)) == \
assoc_laguerre(n, conjugate(alpha), conjugate(x))
pytest.raises(ValueError, lambda: assoc_laguerre(-2.1, alpha, x))
# issue sympy/sympy#10961
X = assoc_laguerre(Rational(5, 2), alpha, x)
assert isinstance(X, assoc_laguerre)
